Finite Group Databasebeta

← #26223679

Subgroups — #26223679

Order 12032.

768 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253303C211yesmaximal, characteristic
2#20252091C211yesmaximal, characteristic
3#20251806C211yesmaximal, characteristic, complemented
4#20244037C211yesmaximal, characteristic, complemented
5#20244037C211yesmaximal, characteristic, complemented
6#20243988C211yesmaximal, characteristic, complemented
7#20243988C211yesmaximal, characteristic, complemented
8#20243688C211yesmaximal, characteristic
9#20243595C211yesmaximal, characteristic, complemented
10#20243488C211yesmaximal, characteristic, complemented
11#20243272C211yesmaximal, characteristic, complemented
12#20243267C211yesmaximal, characteristic, complemented
13#20243232C211yesmaximal, characteristic, complemented
14#20243225C211yesmaximal, characteristic, complemented
15C47×#1753C211yesmaximal, characteristic, complemented
16C2×#13094030C2211yescharacteristic
17C22×#10625329C2211yescharacteristic
18#15689634C2211yescharacteristic
19C2×#13093983C2211yescharacteristic
20#15689465C411yescharacteristic, complemented
21#15689465C411yescharacteristic, complemented
22#15689459C411yescharacteristic, complemented
23#15689454C411yescharacteristic, complemented
24#15689448C411yescharacteristic, complemented
25#15689448C411yescharacteristic, complemented
26#15689442C411yescharacteristic, complemented
27#15689437C411yescharacteristic, complemented
28#15689199—21no—
29#15689199—21no—
30#15689192—21no—
31#15689192—21no—
32C2×#13093960C2211yescharacteristic
33C2×#13093960C2211yescharacteristic
34#15689161C2211yescharacteristic
35#15689161C2211yescharacteristic
36#15689149C2211yescharacteristic
37C2×#13093952C2211yescharacteristic
38C2×#13093952C2211yescharacteristic
39#15689120C2211yescharacteristic
40#15689120C2211yescharacteristic
41#15689089C2211yescharacteristic
42#15689087C2211yescharacteristic
43#15689081—21no—
44#15689081—21no—
45#15689078C2211yescharacteristic
46#15689078C2211yescharacteristic
47#15689078C2211yescharacteristic
48#15689078C2211yescharacteristic
49#15689078C2211yescharacteristic
50M16×Dic47—21no—
51M16×Dic47—21no—
52#15688888C2211yescharacteristic
53#15688888C2211yescharacteristic
54#15688883C2211yescharacteristic
55#15688883C2211yescharacteristic
56#15688883C2211yescharacteristic
57#15688883C2211yescharacteristic
58#15688883C2211yescharacteristic
59#15688883C2211yescharacteristic
60C22×M16×C47C2211yescharacteristic
61C2×C22⋊C8×C47C2211yescharacteristic
62C2×#97×C47C2211yescharacteristic
63C2×C4×C8×C47C2211yescharacteristic
64C4×C22⋊C4×C47C2211yescharacteristic
65C47×#336C2211yescharacteristic
66C47×#336C2211yescharacteristic
67C2×#10625350C2311yescharacteristic
68#13094030—22no—
69#13094030—22no—
70C2×#10625329—21no—
71C2×#10625329—22no—
72C2×#10625329—22no—
73C2×#10625329C2311yescharacteristic
74C2×#10625329C2311yescharacteristic
75C2×#10625329C2311yescharacteristic
76C2×#10625329C2311yescharacteristic
77C2×#10625324—21no—
78C22×C47⋊C8—21no—
79C2×C8×D47—21no—
80#13094007—21no—
81#13094003C2×C411yescharacteristic
82#13094003C2×C411yescharacteristic
83#13093998C2×C411yescharacteristic
84#13093998C2×C411yescharacteristic
85#13093993—21no—
86C4×C22⋊D47—21no—
87C4×C22⋊D47—21no—
88#13093983—21no—
89#13093983—21no—
90#13093983C2×C411yescharacteristic
91#13093983C2×C411yescharacteristic
92#13093983C2311yescharacteristic
93#13093983C2311yescharacteristic
94#13093983C2311yescharacteristic
95C2×C4⋊Dic47C2×C411yescharacteristic
96C2×C4⋊Dic47C2×C411yescharacteristic
97C2×C22⋊Dic47—21no—
98C2×C4×Dic47—21no—
99#13093960—22no—
100#13093960—22no—
101#13093958C2×C411yescharacteristic
102#13093958C2×C411yescharacteristic
103#13093958C2×C411yescharacteristic
104#13093958C2×C411yescharacteristic
105#13093956C2×C411yescharacteristic
106#13093956C2×C411yescharacteristic
107#13093956C2×C411yescharacteristic
108#13093956C2×C411yescharacteristic
109Dic47⋊C8—21no—
110Dic47⋊C8—21no—
111#13093952—22no—
112#13093952—22no—
113#13093949C2×C411yescharacteristic
114#13093949C2×C411yescharacteristic
115#13093949C2×C411yescharacteristic
116#13093949C2×C411yescharacteristic
117#13093946C2×C411yescharacteristic
118#13093946C2×C411yescharacteristic
119#13093946C2×C411yescharacteristic
120#13093946C2×C411yescharacteristic
121C22⋊(C47⋊C8)—21no—
122C22⋊(C47⋊C8)—21no—
123(C2×D47)⋊C8—21no—
124(C2×D47)⋊C8—21no—
125#13093934—21no—
126#13093934—21no—
127#13093934—21no—
128#13093934—21no—
129#13093934—21no—
130C8×Dic47—21no—
131C8×Dic47—21no—
132C8×Dic47—21no—
133C8×Dic47—21no—
134C8×Dic47—21no—
135C23×C4×C47C2311yescharacteristic
136C2×M16×C47—22no—
137C2×M16×C47—22no—
138C2×M16×C47D411yescharacteristic
139C2×M16×C47D411yescharacteristic
140C2×M16×C47D411yescharacteristic
141C2×M16×C47D411yescharacteristic
142C22×C8×C47C2311yescharacteristic
143C22×C8×C47C2311yescharacteristic
144C22×C8×C47C2311yescharacteristic
145C22×C8×C47C2311yescharacteristic
146C2×C22⋊C4×C47—21no—
147C2×C42×C47C2311yescharacteristic
148C2×C42×C47C2311yescharacteristic
149C22⋊C8×C47C2×C411yescharacteristic
150C22⋊C8×C47C2×C411yescharacteristic
151C22⋊C8×C47C2×C411yescharacteristic
152C22⋊C8×C47C2×C411yescharacteristic
153#97×C47—22no—
154C4×C8×C47—22no—
155#95×C47—21no—
156C24×C47#4111yescharacteristic
157C22×C4×C47—21no—
158C22×C4×C47—21no—
159C22×C4×C47—21no—
160C22×C4×C47—21no—
161C22×C4×C47—21no—
162C22×C4×C47—21no—
163C22×C4×C47C22×C411yescharacteristic
164C22×C4×C47C22×C411yescharacteristic
165C22×C4×C47C2×D411yescharacteristic
166C22×C4×C47C2×D411yescharacteristic
167C22×C4×C47#4111yescharacteristic
168C22×C4×C47C2411yescharacteristic
169C2×C22⋊D47—21no—
170C2×C22⋊D47—21no—
171C2×C22⋊D47—21no—
172C22×Dic47—21no—
173C22×Dic47—21no—
174#10625350—21no—
175#10625350—21no—
176#10625350—21no—
177#10625350—21no—
178C2×D188C22×C411yescharacteristic
179C2×C4×D47—21no—
180C2×Dic94C22×C411yescharacteristic
181M16×C47—22no—
182M16×C47—22no—
183M16×C47—22no—
184M16×C47—22no—
185C2×C8×C47—21no—
186C2×C8×C47—21no—
187C2×C8×C47—21no—
188C2×C8×C47—21no—
189C2×C8×C47—21no—
190C2×C8×C47—21no—
191C2×C8×C47C22×C411yescharacteristic
192C2×C8×C47C22×C411yescharacteristic
193C2×C8×C47C22×C411yescharacteristic
194C2×C8×C47C22×C411yescharacteristic
195C2×C8×C47C2×D411yescharacteristic
196C2×C8×C47C2×D411yescharacteristic
197C2×C8×C47C2×D411yescharacteristic
198C2×C8×C47C2×D411yescharacteristic
199C2×C8×C47#4112yes—
200C2×C8×C47#4112yes—
201C22⋊C4×C47—42no—
202C42×C47—21no—
203C42×C47—21no—
204C42×C47—21no—
205C42×C47—21no—
206C22⋊Dic47—21no—
207C22⋊Dic47—21no—
208C22⋊Dic47—21no—
209C22⋊Dic47—21no—
210C22⋊Dic47—21no—
211C22⋊Dic47—21no—
212C22⋊Dic47—21no—
213(C2×D47)⋊C4—21no—
214(C2×D47)⋊C4—21no—
215C4⋊Dic47—21no—
216C4⋊Dic47—21no—
217C4⋊Dic47C22×C411yescharacteristic
218C4⋊Dic47C22×C411yescharacteristic
219C4⋊Dic47C22×C411yescharacteristic
220C4⋊Dic47C22×C411yescharacteristic
221C4⋊Dic47C22×C411yescharacteristic
222C4⋊Dic47C22×C411yescharacteristic
223Dic47⋊C4—21no—
224Dic47⋊C4—21no—
225C4×Dic47—21no—
226C4×Dic47—21no—
227C4×Dic47—21no—
228C4×Dic47—21no—
229C4×Dic47—21no—
230#10625329—24no—
231#10625329—21no—
232#10625329—21no—
233#10625329—21no—
234#10625329—21no—
235#10625329—21no—
236#10625329—21no—
237#10625329#4112yes—
238#10625329#4112yes—
239C2×C47⋊C8—24no—
240C2×C47⋊C8—21no—
241C2×C47⋊C8—21no—
242C2×C47⋊C8—21no—
243C2×C47⋊C8—21no—
244#10625324—42no—
245C8×D47—42no—
246C23×C47—21no—
247C23×C47—21no—
248C23×C47—41no—
249C23×C47—41no—
250C23×C47C4×D411yescharacteristic
251C23×C47C4×D411yescharacteristic
252C23×C47C2×#4111yescharacteristic
253C22×D47—21no—
254C2×C4×C47—21no—
255C2×C4×C47—21no—
256C2×C4×C47—21no—
257C2×C4×C47—21no—
258C2×C4×C47—21no—
259C2×C4×C47—21no—
260C2×C4×C47—21no—
261C2×C4×C47—21no—
262C2×C4×C47—21no—
263C2×C4×C47—21no—
264C2×C4×C47—22no—
265C2×C4×C47—41no—
266C2×C4×C47—41no—
267C2×C4×C47—41no—
268C2×C4×C47—41no—
269C2×C4×C47—41no—
270C2×C4×C47C4×D411yescharacteristic
271C2×C4×C47C4×D411yescharacteristic
272C2×C4×C47C23×C411yescharacteristic
273C2×C4×C47C22×D411yescharacteristic
274C2×C4×C47C2×#4111yescharacteristic
275C2×C4×C47C2×#4111yescharacteristic
276C2×C4×C47C2×#4111yescharacteristic
277C2×C4×C47C2×#4111yescharacteristic
278C22⋊D47—41no—
279C22⋊D47—41no—
280C22⋊D47—81no—
281C22⋊D47—81no—
282C2×Dic47—21no—
283C2×Dic47—21no—
284C2×Dic47—21no—
285C2×Dic47—21no—
286C2×Dic47—21no—
287C2×Dic47—21no—
288C2×Dic47—21no—
289C2×Dic47—81no—
290C2×Dic47—81no—
291D188—21no—
292D188—21no—
293C4×D47—41no—
294C4×D47—41no—
295Dic94—21no—
296Dic94—21no—
297C8×C47—21no—
298C8×C47—21no—
299C8×C47—21no—
300C8×C47—21no—
301C8×C47—21no—
302C8×C47—21no—
303C8×C47#12312yes—
304C8×C47#12312yes—
305C47⋊C8—22no—
306C47⋊C8—22no—
307C47⋊C8—41no—
308C47⋊C8—41no—
309#21594—471nomaximal, Sylow
310C22×C47—21no—
311C22×C47—21no—
312C22×C47—21no—
313C22×C47—21no—
314C22×C47—21no—
315C22×C47—21no—
316C22×C47—21no—
317C22×C47—21no—
318C22×C47—81no—
319C22×C47—81no—
320C22×C47C2×C4×D411yescharacteristic
321C22×C47#52911yescharacteristic
322C22×C47#52911yescharacteristic
323C2×D47—41no—
324C2×D47—41no—
325C2×D47—41no—
326C4×C47—21no—
327C4×C47—21no—
328C4×C47—42no—
329C4×C47—81no—
330C4×C47C4×#4111yescharacteristic
331C4×C47C4×#4111yescharacteristic
332C4×C47C2×#12311yescharacteristic
333C4×C47#52511yescharacteristic
334Dic47—41no—
335Dic47—41no—
336Dic47—41no—
337Dic47—41no—
338Dic47—41no—
339Dic47—41no—
340Dic47—41no—
341#3091—471no—
342#2930—471no—
343#2873—471no—
344#1929—471no—
345#1929—471no—
346#1922—471no—
347#1922—471no—
348#1871—471no—
349#1855—471no—
350#1826—471no—
351#1773—471no—
352#1772—471no—
353#1765—471no—
354#1764—471no—
355#1753D4711yescharacteristic, complemented
356C2×C47—21no—
357C2×C47—21no—
358C2×C47—41no—
359C2×C47—41no—
360C2×C47C4×#12311yescharacteristic
361C2×C47#295611yescharacteristic
362C2×C47#295711yescharacteristic
363D47—81no—
364C2×#131—471no—
365C22×M16—471no—
366C22×M16C2×D4711yescharacteristic
367#525—471no—
368C2×#117—471no—
369#485—471no—
370#485—471no—
371#484—471no—
372#483—471no—
373#482—471no—
374#482—471no—
375#481—471no—
376#480—471no—
377#436—941no—
378#436—941no—
379#435—941no—
380#435—941no—
381C2×#108—471no—
382C2×#108—471no—
383#427—471no—
384#427—471no—
385#424—471no—
386C2×#104—471no—
387C2×#104—471no—
388#416—471no—
389#416—471no—
390#408—471no—
391#407—471no—
392C2×C22⋊C8C2×D4711yescharacteristic
393#405—471no—
394#405—471no—
395#405—471no—
396#405—471no—
397#405—471no—
398#405—941no—
399#405—941no—
400C4×M16—941no—
401C4×M16—941no—
402C2×#97C2×D4711yescharacteristic
403C2×C4×C8C2×D4711yescharacteristic
404C4×C22⋊C4C2×D4711yescharacteristic
405#341—471no—
406#341—471no—
407#339—471no—
408#339—471no—
409#339—471no—
410#339—471no—
411#339—471no—
412#339—471no—
413#336C2×D4711yescharacteristic
414#336C2×D4711yescharacteristic
415C47#2159411yescharacteristic, complemented, Sylow
416C2×#41—471no—
417C23×C4C22×D4711yescharacteristic
418#131—942no—
419#131—942no—
420C2×M16—22no—
421C2×M16—22no—
422C2×M16—471no—
423C2×M16—471no—
424C2×M16—471no—
425C2×M16—471no—
426C2×M16—942no—
427C2×M16—942no—
428C2×M16—941no—
429C2×M16—941no—
430C2×M16D18811yescharacteristic
431C2×M16D18811yescharacteristic
432C2×M16D18811yescharacteristic
433C2×M16D18811yescharacteristic
434C22×C8—941no—
435C22×C8—941no—
436C22×C8C22×D4711yescharacteristic
437C22×C8C22×D4711yescharacteristic
438C22×C8C22×D4711yescharacteristic
439C22×C8C22×D4711yescharacteristic
440#123—941no—
441C22⋊Q8—471no—
442C22⋊Q8—471no—
443#121—471no—
444#121—471no—
445#120—941no—
446C4×D4—941no—
447C4×D4—941no—
448#117—471no—
449#117—471no—
450#117—471no—
451#117—471no—
452#117—471no—
453#117—941no—
454#117—941no—
455C2×C4⋊C4—471no—
456C2×C4⋊C4—471no—
457C2×C22⋊C4—21no—
458C2×C22⋊C4—941no—
459C2×C42—941no—
460C2×C42C22×D4711yescharacteristic
461C2×C42C22×D4711yescharacteristic
462#108—942no—
463#108—942no—
464#107—471no—
465#107—471no—
466#107—471no—
467#107—471no—
468#106—471no—
469#106—471no—
470#106—471no—
471#106—471no—
472C4⋊C8—941no—
473C4⋊C8—941no—
474#104—942no—
475#104—942no—
476#103—471no—
477#103—471no—
478#103—471no—
479#103—471no—
480#102—471no—
481#102—471no—
482#102—471no—
483#102—471no—
484C22⋊C8—941no—
485C22⋊C8—941no—
486C22⋊C8—941no—
487C22⋊C8—941no—
488C22⋊C8C4×D4711yescharacteristic
489C22⋊C8C4×D4711yescharacteristic
490C22⋊C8C4×D4711yescharacteristic
491C22⋊C8C4×D4711yescharacteristic
492#97—22no—
493#97—941no—
494#97—941no—
495#97—941no—
496#97—941no—
497#97—941no—
498C4×C8—22no—
499C4×C8—941no—
500C4×C8—941no—
501C4×C8—941no—
502C4×C8—941no—
503C4×C8—941no—
504#95—21no—
505C24#1062535011yescharacteristic
506#41—941no—
507#41—941no—
508#41—941no—
509#41—941no—
510C2×Q8—471no—
511C2×D4—471no—
512C2×D4—941no—
513C2×D4—941no—
514C2×D4—941no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—941no—
522C22×C4—941no—
523C22×C4—941no—
524C22×C4C2×C4×D4711yescharacteristic
525C22×C4C2×C4×D4711yescharacteristic
526C22×C4C2×D18811yescharacteristic
527C22×C4C2×D18811yescharacteristic
528C22×C4#1062535011yescharacteristic
529C22×C4C23×D4711yescharacteristic
530M16—22no—
531M16—22no—
532M16—22no—
533M16—22no—
534M16—472no—
535M16—472no—
536M16—944no—
537M16—941no—
538M16—941no—
539M16—941no—
540M16—941no—
541M16—941no—
542M16—941no—
543M16—1882no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—944no—
551C2×C8—941no—
552C2×C8—941no—
553C2×C8—941no—
554C2×C8—941no—
555C2×C8—1882no—
556C2×C8C2×C4×D4711yescharacteristic
557C2×C8C2×C4×D4711yescharacteristic
558C2×C8C2×C4×D4711yescharacteristic
559C2×C8C2×C4×D4711yescharacteristic
560C2×C8C2×D18811yescharacteristic
561C2×C8C2×D18811yescharacteristic
562C2×C8C2×D18811yescharacteristic
563C2×C8C2×D18811yescharacteristic
564C2×C8#1062535212yes—
565C2×C8#1062535212yes—
566C4⋊C4—471no—
567C4⋊C4—471no—
568C4⋊C4—471no—
569C4⋊C4—471no—
570C4⋊C4—471no—
571C4⋊C4—471no—
572C4⋊C4—941no—
573C4⋊C4—941no—
574C4⋊C4—941no—
575C4⋊C4—941no—
576C22⋊C4—42no—
577C22⋊C4—941no—
578C22⋊C4—941no—
579C22⋊C4—941no—
580C22⋊C4—941no—
581C22⋊C4—941no—
582C22⋊C4—941no—
583C22⋊C4—941no—
584C22⋊C4—941no—
585C22⋊C4—941no—
586C42—21no—
587C42—21no—
588C42—21no—
589C42—21no—
590C42—941no—
591C42—941no—
592C42—941no—
593C42—941no—
594C42—941no—
595C23—21no—
596C23—21no—
597C23—41no—
598C23—41no—
599C23—941no—
600C23C4×D18811yescharacteristic
601C23C4×D18811yescharacteristic
602C23C2×#1062535011yescharacteristic
603Q8—941no—
604Q8—941no—
605D4—941no—
606D4—941no—
607D4—3761no—
608D4—3761no—
609D4—1881no—
610D4—1881no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—941no—
628C2×C4—941no—
629C2×C4—941no—
630C2×C4—941no—
631C2×C4—941no—
632C2×C4—941no—
633C2×C4—941no—
634C2×C4—3761no—
635C2×C4—3761no—
636C2×C4—1881no—
637C2×C4—1881no—
638C2×C4C4×D18811yescharacteristic
639C2×C4C4×D18811yescharacteristic
640C2×C4C22×C4×D4711yescharacteristic
641C2×C4C22×D18811yescharacteristic
642C2×C4#41×D4711yescharacteristic
643C2×C4#41×D4711yescharacteristic
644C2×C4C2×#1062535211yescharacteristic
645C2×C4C2×#1062535211yescharacteristic
646C8—21no—
647C8—21no—
648C8—21no—
649C8—21no—
650C8—21no—
651C8—21no—
652C8—942no—
653C8—942no—
654C8—1881no—
655C8—1881no—
656C8#1309400812yes—
657C8#1309400812yes—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—81no—
667C22—81no—
668C22—1881no—
669C22—1881no—
670C22—1881no—
671C22C2×C4×D18811yescharacteristic
672C22#1568965311yescharacteristic
673C22#1568965311yescharacteristic
674C4—21no—
675C4—21no—
676C4—42no—
677C4—81no—
678C4—1881no—
679C4—1881no—
680C4—1881no—
681C4—1881no—
682C4—1881no—
683C4—1881no—
684C4—1881no—
685C4C4×#1062535211yescharacteristic
686C4C4×#1062535211yescharacteristic
687C4C2×#1309400811yescharacteristic
688C4#1568963511yescharacteristic
689C2—21no—
690C2—21no—
691C2—41no—
692C2—41no—
693C2—3761no—
694C2C4×#1309400811yescharacteristic
695C2#2025229911yescharacteristic
696C2#2025230611yescharacteristic
subgroupG/H
centerC2×C4C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1753C2
socleC22×C47C2×C4×D4
prime radical#15689634C22
cosocleC22×C4×C47C24
hypercenter#1753D47
hypocenterC47#21594
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×M16 ◁ #1753

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20251806C2semidirect
4. #20244037C2semidirect
5. #20244037C2semidirect
6. #20243988C2semidirect
7. #20243988C2semidirect
9. #20243595C2semidirect
10. #20243488C2semidirect
11. #20243272C2semidirect
12. #20243267C2semidirect
13. #20243232C2semidirect
14. #20243225C2semidirect
15. C47×#1753C2semidirect
20. #15689465C4semidirect
21. #15689465C4semidirect
22. #15689459C4semidirect
23. #15689454C4semidirect
24. #15689448C4semidirect
25. #15689448C4semidirect
26. #15689442C4semidirect
27. #15689437C4semidirect
355. #1753D47semidirect
415. C47#21594semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202533032
#202520912
#202518062
#202440372
#202439882
#202436882
#202435952
#202434882
#202432722
#202432672
#202432322
#202432252
C47×#17532
#2159447